Question

What is the minimum number of atoms that could be contained in the unit cell of an element with a face-centered cubic lattice?

Short answer

The minimum number of atoms in the unit cell of an element with a face-centered cubic lattice is 4. This includes 1 atom contributed from the 8 corners of the cube and 3 atoms contributed from the face centers of the cube.

Step-by-step solution

Understanding the face-centered cubic lattice structure

A face-centered cubic (FCC) lattice structure is a type of crystalline structure where atoms are located at each corner of the cube and also at the center of each face of the cubic unit cell.

Counting atoms at the corners

There are 8 corners in a cubic unit cell, and each corner is shared by 8 adjacent unit cells. Therefore, each corner contributes \(\frac{1}{8}\) of an atom to a unit cell. So, for all the 8 corners combined, the contribution of atoms to the unit cell from the corners is: \( 8 \times \frac{1}{8} = 1 \)

Counting atoms at the face centers

There are 6 faces in a cubic unit cell, and each face-center is shared by 2 adjacent unit cells. Therefore, each face-center contributes \(\frac{1}{2}\) of an atom to a unit cell. So, for all the 6 face-centers combined, the contribution of atoms to the unit cell from the face centers is: \( 6 \times \frac{1}{2} = 3 \)

Calculating the total number of atoms in the unit cell

Now we just add up the contributions from the corners and the face centers to find the total number of atoms in the unit cell: \( 1 (from \:corners) + 3(from\:face-centers)= 4 \) Thus, the minimum number of atoms contained in the unit cell of an element with a face-centered cubic lattice is 4.

Key concepts

Unit Cell

The unit cell is the basic structural component of a crystalline solid. It is the smallest repeating unit that represents the overall structure. Imagine it as a tiny block that, when stacked in three-dimensional space, forms the entire lattice. Each unit cell is defined by a specific arrangement of atoms, which is crucial for the material's properties. In crystalline solids, these unit cells repeat in a consistent pattern, creating the solid's external shape. The shape and content of a unit cell determine the material's density, packing efficiency, and even electrical properties. Understanding the unit cell is like having the key to unlock the material's full characteristics.

Crystalline Structure

Crystalline structures are solids where the atoms are arranged in a highly ordered manner. Unlike amorphous solids, which lack a repeating pattern, crystalline structures have atoms systematically positioned throughout the material. This orderliness leads to the formation of distinct patterns and shapes in these materials.

• Symmetry: A key feature of crystalline structures is their symmetry. This symmetry can range from simple cubic to more complex hexagonal arrangements.
• Repetition: The structured repetition of atoms in a crystal causes unique physical properties, such as the ability to split into clean, flat planes known as cleavage planes.

These ordered networks are what give crystals their firmness, clarity, and sometimes even their optical properties, like in diamonds.

Atoms in a Lattice

In a crystalline lattice, atoms are arranged in a repetitive geometric pattern. It's important to understand that each atom within the lattice interacts with its neighbors. These interactions contribute to characteristics such as strength, flexibility, and thermal stability. The positioning of atoms affects not only the macroscopic appearance of the material but also its microscopic properties.

• Coordination Number: This is the number of nearest neighboring atoms surrounding an atom in a lattice. It affects the material's stability.
• Interatomic Distances: The distances between atoms can affect the overall compactness and density of the material.

Understanding the arrangement of atoms in a lattice helps predict and explain the physical properties of materials.

Face-Centered Cubic Structure

The face-centered cubic (FCC) structure is a specific arrangement of atoms within a crystalline solid. In this structure, you will find

• Atoms at each of the eight corners of a cube, contributing \(\frac{1}{8}\) of an atom each to the unit cell.
• Atoms at the center of each of the six faces, each contributing \(\frac{1}{2}\) of an atom to the unit cell.

The face-centered cubic structure is known for its high packing efficiency, meaning that the atoms are closely packed, utilizing space effectively. The total contribution from corners and face centers in the FCC structure sums to a minimum of 4 atoms per unit cell. The FCC lattice is common in metals such as aluminum, copper, and gold. Its dense arrangement contributes to the sturdiness and higher density of these materials.